For the last 150 years, riflemen have attempted to predict the paths of bullets fired over a long range. The discovery that the bullet path is parabolic when the effects of air resistance are ignored was derived much earlier by Galileo. However, the computation becomes considerably more complicated when the slowing of the bullet caused by air resistance is considered. Air resistance is critical to consider in order to obtain an accurate prediction because air resistance is the primary force acting on a projectile. For modern rifle bullets, the force of air resistance can be 50 to 100 times stronger than the force of gravity. Newton postulated that the retarding force or drag on a bullet caused by air was proportional to the square of the bullet's velocity. A similar approach continues to be used by modern ballisticians.
Starting around 1850, Newton's basic idea was refined to the formula F∝G(v)v2/C, where F is the retarding force on the bullet, ∝ is read as “is proportional to,” G(v) is the “drag coefficient” defining the drag of the “standard bullet” as a function of velocity, v is the velocity of the bullet, and C is the “ballistic coefficient” or drag of the standard bullet divided by the drag of the tested bullet at a given velocity.
Predictions of exterior ballistics are made using the above equation starting at some initial velocity V0 and integrating the acceleration (force) as a function of time to find velocity as a function of time. A second integration provides distance as a function of time. The initial velocity V0 is often assumed from published ammunition specifications. It is also assumed that G(v) is adequately represented by one of the standard drag functions (typically G1). It is further assumed that the ballistic coefficient C of the bullet is accurately known.
For 150 years, ballisticians have started with this set of assumptions and have made predictions of downrange performance. Their predictions have been computational extrapolations from characteristics measured near the muzzle and are based on the assumed drag function, estimated ballistic coefficient, and estimated initial velocity. These predictions have usually been sufficiently accurate to “get on paper” (hitting a portion of the target) at long range and the shooter expects to “come up or down a few clicks” to refine his sighting. There has been little effort to refine the predictions with actual long range tests. There has been no formal procedure to assure agreement between predictions and measured results at intermediate ranges. Mathematically, the ballisticians have performed two integrations, but have not applied a terminal value or condition to the results of the integration.
There are three primary sources of error in the predictions: the initial velocity V0 can vary significantly from gun to gun, the assumed drag function G(v) does not exactly fit the bullet, and the assumed ballistic coefficient can vary with velocity and from gun to gun.
It is instructive to look at a graph of bullet travel (distance) versus time. Referring now to FIG. 1, the x-axis shows the length of time that has passed since the bullet was discharged from the gun, and the y-axis shows the distance the bullet has traveled from the gun. The initial slope of the curve represents the initial velocity V0 of the bullet, and the slope at any point represents the velocity at that time. The flattening of the curve indicates the velocity reduction that occurs as the bullet is slowed by air drag. The degree of curvature indicates the drag as velocity diminishes. The point shown near the end of the curve represents the cumulative effects of the drag applied to the initial velocity. This point characterizes the cumulative long range performance of the bullet, but has historically been neglected because it is difficult to measure. The current invention illustrates both the requirement for the measurement and a preferred method of measurement.
To make calculated bullet path predictions match reality, two conditions are essential. First, the initial slope must match the measured initial velocity. Second, the curve predicted from the assumed drag function and ballistic coefficient must pass through the measured long time and long distance point where the bullet impacts the target.
Better bullet path predictions are required by snipers and others wanting a high probability of a first round hit at long range. In recent years, snipers have been trained by Todd Hodnett, a well-known long-range shooting instructor, to use a procedure in which the initial velocity used in their predictions is “trued” or arbitrarily changed to make their predictions agree with actual shooting results at a specified long distance. This procedure provides a correction that works to accurately predict a bullet path at the specific distance measured. However, snipers need bullet path predictions for their firearms that are accurate at all distances.
Therefore, a need exists for a new and improved system for measuring exterior ballistics that measures initial velocity and time of flight of a bullet to a known distance, calculates a ballistic coefficient for the bullet, and enhances the ability of traditional predictive equations and procedures to accurately predict bullet paths for other distances and conditions. In this regard, the various embodiments of the present invention substantially fulfill at least some of these needs. In this respect, the system for measuring exterior ballistics according to the present invention substantially departs from the conventional concepts and designs of the prior art, and in doing so provides an apparatus primarily developed for the purpose of providing a system that measures initial velocity and time of flight of a bullet to a known distance, calculates a ballistic coefficient for the bullet, and enhances the ability of traditional predictive equations and procedures to accurately predict bullet paths for other distances and conditions.
Prior art Doppler radar systems are available that are capable of tracking a rifle bullet over long range and provide the same information as is measured by the current invention. The Doppler data is sufficiently complete that it yields a unique G(v) drag coefficient exactly fitting the bullet under test. Hence the “ballistic coefficient” C becomes unity. However, such Doppler systems are very expensive, they are not man-portable, and they require significantly more support.